The Annals of Probability

A stochastic Burgers equation from a class of microscopic interactions

Patrícia Gonçalves, Milton Jara, and Sunder Sethuraman

Full-text: Open access

Abstract

We consider a class of nearest-neighbor weakly asymmetric mass conservative particle systems evolving on $\mathbb{Z}$, which includes zero-range and types of exclusion processes, starting from a perturbation of a stationary state. When the weak asymmetry is of order $O(n^{-\gamma})$ for $1/2<\gamma\leq1$, we show that the scaling limit of the fluctuation field, as seen across process characteristics, is a generalized Ornstein–Uhlenbeck process. However, at the critical weak asymmetry when $\gamma=1/2$, we show that all limit points satisfy a martingale formulation which may be interpreted in terms of a stochastic Burgers equation derived from taking the gradient of the KPZ equation. The proofs make use of a sharp “Boltzmann–Gibbs” estimate which improves on earlier bounds.

Article information

Source
Ann. Probab., Volume 43, Number 1 (2015), 286-338.

Dates
First available in Project Euclid: 12 November 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1415801558

Digital Object Identifier
doi:10.1214/13-AOP878

Mathematical Reviews number (MathSciNet)
MR3298474

Zentralblatt MATH identifier
1311.60069

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C22: Interacting particle systems [See also 60K35]

Keywords
KPZ equation Burgers weakly asymetric zero-range kinetically constrained speed-change fluctuations

Citation

Gonçalves, Patrícia; Jara, Milton; Sethuraman, Sunder. A stochastic Burgers equation from a class of microscopic interactions. Ann. Probab. 43 (2015), no. 1, 286--338. doi:10.1214/13-AOP878. https://projecteuclid.org/euclid.aop/1415801558


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